Theory and Methods of Statistics

11.10.4 Estimation of Variance Components

For the model given in Eq. (7)

$\begin{array}{ll}\hfill \text{E}\left[\mathit{Y}\right]& =\mathit{X}\mathit{\beta },\text{and}\hfill \\ \hfill \mathit{\Sigma }& =\text{Cov}\left[\mathit{Y}\right]={\sigma }_1^2{\mathit{Z}}_1{\mathit{Z}}_1^T+\cdots +{\sigma }_r^2{\mathit{Z}}_r{\mathit{Z}}_r^T+{\sigma }_0^{2}I.\hfill \end{array}$

Note that we have changed the notations a bit and now σ² is denoted by ${\sigma }_0^2$. We now focus on the problem of estimating the variance components ${\sigma }_0^2,{\sigma }_1^2,\dots ,{\sigma }_r^2$.

There are a number of well-known methods for estimating the variance components including

(i) Henderson’s method III,

(ii) Maximum likelihood,

(iii) Restricted maximum likelihood ...